HETEROSIS IN POPULATION GENETICS 



157 



orclinates were obtained by taking the square root of the observed frequencies 

 of homozygous recessives. 



The following conclusions can be drawn: (1) the three experimental popu- 

 lations, each being run in duplicate, have reached the same gene frequency 

 at about the .579 point; (2) natural selection has been acting on the three 

 populations producing the same end results, irrespective of the initial gene 

 frequency; (3) natural selection has been acting in favor of the heterozygous 

 flies; and (4) the homozygous mutant seems to be slightly superior in its 

 survival value to the homozygous normal allele. 



It was of considerable interest to determine whether the intensity of selec- 

 tion operating in the three experiments was the same. Since the three experi- 

 mental curves (each being the mean of the two duplicate populations) could 

 not be compared directly. Dr. L. L. Cavalli elaborated a mathematical 

 analysis of the problem (Cavalli, 1950). The function of gene frequency linear 

 with time F, when the heterozygote is at an advantage, is given by: 



Y = Qe^Og p-\-pe\0g q -\0g[pe-p] , 



where p and q are the gene frequencies at the beginning of the experiment in 

 a random breeding population, and pe and qe are the equilibrium frequencies. 

 By means of this function it is possible to transform the experimental curves 

 to linear ones. Results can then be plotted graphically for the three experi- 

 ments. Fitting straight lines with the method of maximum likelihood, one 

 obtains the following values for the constants of the linear regression equa- 

 tion: 



The position is the transformed value of the initial gene frequency which 

 is given in the last column, and is in good agreement with the experimental 

 value. If one tests the parallelism of the three regression lines so obtained, one 

 gets a chi square of 4.0 with two degrees of freedom. Parallelism therefore 

 seems to be satisfactory. This implies that the intensity of selection is inde- 

 pendent of initial conditions. 



If we take these results together with the two independent occurrences of 

 the same mutant gene in different genotypical milieus, it seems safe to main- 

 tain that such a gene has a positive selective value with respect to its normal 

 allele, and that selection is acting mainly through a typical heterosis mecha- 

 nism. It is worth while to stress that this gene was found both in natural and 



